Source: Comprehensive Guide to Nuclear Physics, Atomic Structure, and Quantum Mechanics (Purdue University)
Tags: Bohr model, quantised orbits, hydrogen atom, energy levels, electron transitions, Schrodinger equation, wavefunction, probability density, atomic orbitals, quantum numbers, principal quantum number, angular momentum quantum number, magnetic quantum number, spin quantum number, Pauli exclusion principle, Aufbau principle, Hund's rule, electron configuration
Difficulty: Intermediate Prerequisites: Electromagnetic radiation basics (photon energy, E = hν). Familiarity with the photoelectric effect and atomic line spectra from the previous notes.
This is where atomic physics shifts from "what does the nucleus do" to "what do the electrons do." The Bohr model was the first attempt to explain why atoms emit light at specific wavelengths, and it works well for hydrogen. Quantum mechanics replaced it with a far more powerful framework: wavefunctions, probability densities, and four quantum numbers that together describe every electron in every atom. Electron configuration rules (Aufbau, Pauli exclusion, Hund's rule) determine how electrons fill orbitals, which in turn determines an element's chemical behaviour. If you understand this material, you understand why the periodic table is shaped the way it is.
The Bohr model describes electrons in discrete orbits with quantised energies, but only works for hydrogen. Quantum mechanics replaces fixed orbits with wavefunctions (solutions to the Schrodinger equation) whose squares give the probability of finding an electron in a given region. Four quantum numbers (n, l, m_l, m_s) specify every electron's state, and the Aufbau principle, Pauli exclusion principle, and Hund's rule govern the order in which orbitals are filled.
Bohr model
A model of the atom where electrons orbit the nucleus in fixed, quantised energy levels (n = 1, 2, 3, ...). It explains hydrogen's line spectrum but fails for multi-electron atoms. Think of it as the "planetary model with rules."
Stationary state
An allowed energy level in the Bohr model. An electron in a stationary state does not radiate energy; it only emits or absorbs a photon when it transitions between states.
Electron transition
The movement of an electron from one energy level to another. A jump to a higher level absorbs a photon; a drop to a lower level emits one. The photon's energy equals the energy difference between the two levels.
Wavefunction (ψ)
A mathematical function that describes the quantum state of an electron. It is a solution to the Schrodinger equation. The wavefunction itself does not have a direct physical meaning, but its square does.
Probability density (|ψ|²)
The square of the wavefunction at a given point. It tells you the probability of finding the electron at that location. In simple terms, where |ψ|² is large, the electron is likely to be found.
Atomic orbital
A region of space where there is a high probability of finding an electron. Orbitals are the shapes you see in textbooks (spheres for s, dumbbells for p, cloverleaves for d).
Principal quantum number (n)
Specifies the energy level or shell. Takes positive integer values (1, 2, 3, ...). Higher n means higher energy and larger average distance from the nucleus.
Angular momentum quantum number (l)
Specifies the shape of the orbital. Ranges from 0 to n − 1. Values correspond to orbital types: l = 0 (s), l = 1 (p), l = 2 (d), l = 3 (f).
Magnetic quantum number (m_l)
Specifies the orientation of the orbital in space. Ranges from −l to +l. For example, a p orbital (l = 1) has three orientations: m_l = −1, 0, +1.
Spin quantum number (m_s)
Specifies the electron's intrinsic spin. Only two values: +½ or −½. Two electrons in the same orbital must have opposite spins.
Pauli exclusion principle
No two electrons in the same atom can share the same set of four quantum numbers. This limits each orbital to a maximum of two electrons, and those two must have opposite spins.
Aufbau principle
Electrons fill orbitals starting from the lowest available energy level. "Aufbau" is German for "building up."
Hund's rule
When filling orbitals of the same energy (degenerate orbitals), electrons occupy them singly with parallel spins before pairing up. Think of it as "one electron per seat before anyone doubles up."
Bohr proposed that electrons orbit the nucleus at specific, fixed distances (quantised radii), each corresponding to a principal quantum number n.
An electron in orbit n has a definite energy. For a hydrogen-like atom (one electron, nuclear charge Z):
E_n = −2.18 × 10⁻¹⁸ × (Z² / n²) J
Transitions between levels produce or absorb photons with energy ΔE = hν.
When an electron drops from a higher n to a lower n, a photon is emitted.
When an electron absorbs a photon of exactly the right energy, it jumps to a higher n.
This neatly explains hydrogen's discrete line spectrum: each spectral line corresponds to a specific transition between energy levels.
Limitations of the Bohr model:
Works only for hydrogen and hydrogen-like ions (single-electron species such as He⁺ or Li²⁺).
Fails for multi-electron atoms because it cannot account for electron-electron repulsion.
Treats electrons as particles in fixed circular orbits. In reality, electrons behave as waves, and their positions are described probabilistically.
The Schrodinger equation (Hψ = Eψ) is the foundation of quantum mechanics. Its solutions, the wavefunctions (ψ), describe the allowed states of electrons in atoms.
The wavefunction itself can be positive, negative, or complex. Its physical meaning comes from |ψ|², the probability density, which is always positive and tells you where the electron is likely to be found.
Atomic orbitals are the visual representation of these probability distributions. Each orbital has a characteristic shape determined by the quantum numbers.
Orbital shapes by l value:
l = 0 (s orbitals): Spherically symmetric. One orientation per shell.
l = 1 (p orbitals): Dumbbell-shaped along an axis. Three orientations per shell (p_x, p_y, p_z).
l = 2 (d orbitals): Cloverleaf or similar shapes. Five orientations per shell.
l = 3 (f orbitals): Complex multi-lobed shapes. Seven orientations per shell.
Each electron in an atom is described by four quantum numbers. Together, they are like an address: n is the floor, l is the wing, m_l is the room, and m_s is which bed in the room.
n (principal): 1, 2, 3, ... Determines the shell and energy level. The number of orbitals in shell n is n².
l (angular momentum): 0 to n − 1. Determines the subshell and orbital shape.
In shell n = 3, l can be 0, 1, or 2 (s, p, d subshells).
m_l (magnetic): −l to +l. Determines which specific orbital within the subshell.
For l = 2 (d subshell), m_l = −2, −1, 0, +1, +2 (five d orbitals).
m_s (spin): +½ or −½. Each orbital holds at most two electrons with opposite spins.
Aufbau principle: Fill from the lowest energy orbital upward. The filling order is:
1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p
Note that 4s fills before 3d, and 5s fills before 4d. This is because of the way orbital energies overlap in multi-electron atoms.
Pauli exclusion principle: Each orbital can hold at most two electrons, which must have opposite spins (+½ and −½).
Hund's rule: Within a subshell, place one electron in each orbital (all with the same spin) before pairing any. For example, the three 2p orbitals each get one electron before any of them gets a second.
Example: Oxygen (Z = 8)
Configuration: 1s² 2s² 2p⁴
The 1s and 2s orbitals are full (two electrons each). The three 2p orbitals contain four electrons total: each of the three orbitals gets one electron first (with parallel spins), then the fourth electron pairs in one of them.
Example: Iron (Z = 26)
Configuration: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶
Note that 4s is listed before 3d in the configuration because 4s fills first, even though 3d has a lower principal quantum number.
Bohr model energy (hydrogen-like atoms): E_n = −2.18 × 10⁻¹⁸ × (Z² / n²) J
Energy of a transition: ΔE = E_final − E_initial = hν
Schrodinger equation: Hψ = Eψ
Number of orbitals in shell n: n² orbitals
Maximum electrons in shell n: 2n² electrons
Maximum electrons in a subshell: 2(2l + 1) electrons
Electron configurations determine chemical reactivity, which is why the periodic table is organised the way it is. Elements in the same group have similar valence electron configurations and therefore similar chemical behaviour. The Bohr model's energy-level transitions are the basis of spectroscopy, used to identify elements in everything from forensic samples to the atmospheres of exoplanets. Lasers work by stimulating specific electron transitions to produce coherent light at a precise wavelength.
Students often think the Bohr model is "wrong." It is incomplete rather than wrong: it gives correct energy levels for hydrogen and remains a useful stepping stone. The quantum mechanical model supersedes it but does not erase its results for single-electron systems.
A common mistake is treating orbitals as fixed paths or tracks. An orbital is a probability distribution, not a trajectory. The electron is not circling the nucleus like a planet.
Many students confuse shell (n) and subshell (l). Shell 3 contains three subshells (3s, 3p, 3d), not three orbitals. The number of orbitals within shell 3 is 3² = 9.
Electron configuration errors often come from forgetting that 4s fills before 3d. Use the diagonal rule or the Aufbau filling order to avoid this.
⚠️ Be prepared to write electron configurations for any element up to Z ≈ 36 (krypton). Know the filling order cold.
⚠️ Exam questions frequently ask for the set of four quantum numbers for a specific electron. Remember: n defines the range of l, l defines the range of m_l, and m_s is always +½ or −½.
⚠️ Hund's rule is commonly tested with orbital diagrams. If asked to draw the orbital diagram for nitrogen (1s² 2s² 2p³), each of the three 2p orbitals gets one electron with the same spin direction, not two in one and one in another.
⚠️ The Bohr energy equation (E_n = −2.18 × 10⁻¹⁸ Z²/n²) is tested for hydrogen (Z = 1). You may be asked to calculate the wavelength of light emitted when an electron drops from n = 4 to n = 2 (the visible Balmer series).
⚠️ Know the Pauli exclusion principle as a limit: it caps each orbital at two electrons. If a question asks for the maximum number of electrons in a shell or subshell, use 2n² or 2(2l + 1).
True or false: The Bohr model accurately predicts the spectrum of helium (a two-electron atom).
Fill in the blank: The angular momentum quantum number l = 2 corresponds to a ___ subshell.
True or false: Two electrons in the same orbital can have the same spin quantum number.
Fill in the blank: The maximum number of electrons that can occupy the n = 3 shell is ___.
True or false: According to Hund's rule, electrons pair up in an orbital before filling empty orbitals of the same energy.
Answers: 1. False (Bohr works only for one-electron species). 2. d. 3. False (Pauli exclusion requires opposite spins). 4. 18 (2 × 3² = 18). 5. False (they fill singly first, then pair).
Q: Calculate the energy of a photon emitted when an electron in a hydrogen atom transitions from n = 3 to n = 1.
A: E₃ = −2.18 × 10⁻¹⁸ / 9 = −2.42 × 10⁻¹⁹ J. E₁ = −2.18 × 10⁻¹⁸ / 1 = −2.18 × 10⁻¹⁸ J. ΔE = E₁ − E₃ = −2.18 × 10⁻¹⁸ − (−2.42 × 10⁻¹⁹) = −1.94 × 10⁻¹⁸ J. The photon carries 1.94 × 10⁻¹⁸ J of energy (the sign indicates emission).
Q: What are the four quantum numbers for one of the 3d electrons?
A: n = 3, l = 2, m_l could be any value from −2 to +2 (say, m_l = 0), m_s = +½ or −½ (say, +½). So one valid set is (3, 2, 0, +½).
Q: Write the electron configuration for chlorine (Z = 17).
A: 1s² 2s² 2p⁶ 3s² 3p⁵.
Q: Why does the Bohr model fail for lithium (Z = 3)?
A: Lithium has three electrons. The Bohr model cannot account for electron-electron repulsion between multiple electrons, so its single-electron energy formula does not give correct energy levels for lithium or any multi-electron atom.
Q: Explain why 4s fills before 3d in multi-electron atoms, even though n = 3 is a lower shell than n = 4.
A: In multi-electron atoms, electron-electron repulsion and shielding effects alter the relative energies of subshells. The 4s orbital penetrates closer to the nucleus and experiences less shielding than 3d, giving it a slightly lower effective energy. This is why the Aufbau order places 4s before 3d.
The Bohr model connects back to the atomic line spectra discussed in the electromagnetic radiation notes: each spectral line is a specific Bohr transition. Quantum numbers and electron configurations are the foundation for understanding chemical bonding (ionic and covalent), periodic trends (ionisation energy, electron affinity, electronegativity), and molecular orbital theory, all of which build directly on the material here.
Bohr model, hydrogen atom energy levels, quantised orbits, electron transitions, Balmer series, Lyman series, Schrodinger equation, wavefunction, psi, probability density, atomic orbital, s orbital, p orbital, d orbital, f orbital, principal quantum number, angular momentum quantum number, magnetic quantum number, spin quantum number, Pauli exclusion principle, Aufbau principle, Hund's rule, electron configuration, orbital diagram, orbital filling order, degenerate orbitals, shell, subshell, quantum mechanics, wave-particle duality